Area of Tangent on a circle

Theoretically it is said that, tangent touches to a single point on a circle. But If my circle is very big, and large enough, then i think, it should not be a just single point where my tangent is touching, though is will be a very small portion depending on how large is the circle.

If i have a perfect sphere of size of earth, then a perfectly flat surface of size of football field will completely be touching on to the earth's surface, and is not just at one point!

So, my question is, how big should be the radius of a circle, to perfectly allow 1 meter of area of a tangent touching perfectly on it?

from a topological point of view, both structures are infinite; without beginning or end. ie the "endpoints" of a line can coincide at infinity, and thus form a closed loop topologically equivalent to a circle. Furthermore, any segment of the line WILL contain the "center" of the line. I therefore propose that this center is the intersection of the endpoints of the line

that is why I used quotes, they are not really points that terminate the line. they are more like the boundary of infinity; two coincident lines can "grow" at different rates, and the line that grows fastest will enclose the other line. The enclosed line would have endpoints within the outer line, as it is entirely contained in the outer line.